"Operads" are mathematical devices which model many sorts of algebras (such as associative, commutative, Lie, Poisson, alternative, Leibniz, etc., including those defined up to homotopy, such as \(A_{\infty}\)-algebras). Since the notion of an operad appeared in the seventies in algebraic topology, there has been a renaissance of this theory due to the discovery of relationships with graph cohomology, Koszul duality, representation theory, combinatorics, cyclic cohomology, moduli spaces, knot theory, and quantum field theory.

This renaissance was recognized at a special session "Moduli Spaces, Operads, and Representation Theory" of the AMS meeting in Hartford, CT (March 1995), and at a conference "Opérades et Algèbre Homotopique" held at the Centre International de Rencontres Mathématiques at Luminy, France (May-June 1995). Both meetings drew a diverse group of researchers.

The authors have arranged the contributions so as to emphasize certain themes around which the renaissance of operads took place: homotopy algebra, algebraic topology, polyhedra and combinatorics, and applications to physics.

Readership

Graduate students, research mathematicians, mathematical physicists, and physicists interested in general algebraic systems.

Table of Contents

• J. P. May -- Definitions: operads, algebras, and modules
• J. Stasheff -- The pre-history of operads
• J. P. May -- Operads, algebras, and modules
• A. Tonks -- Relating the associahedron and the permutohedron
• C. Berger -- Combinatorial models for real configuration spaces and \(E_n\)-operads
• J. Stasheff -- From operads to `physically' inspired theories
• A. V. Gnedbaye -- Opérades des algèbres \((k+1)\)-aires
• J.-M. Oudom -- Coproduct and cogroups in the category of graded dual Leibniz algebras
• H.-J. Baues, M. Jibladze, and A. Tonks -- Cohomology of monoids in monoidal categories
• T. F. Fox and M. Markl -- Distributive laws, bialgebras, and cohomology
• D. Balavoine -- Deformations of algebras over a quadratic operad
• T. P. Bisson and A. Joyal -- \(Q\)-rings and the homology of the symmetric groups
• J. P. May -- Operadic tensor products and smash products
• T. Kimura, A. A. Voronov, and G. J. Zuckerman -- Homotopy Gerstenhaber algebras and topological field theory
• Y.-Z. Huang -- Intertwining operator algebras, genus-zero modular functors, and genus-zero conformal field theories
• B. Feigin and F. Malikov -- Modular functor and representation theory of \(\widehat {sl_2}\) at a rational level
• J. Morava -- Quantum generalized cohomology
• J.-L. Brylinski and D. A. McLaughlin -- Non-commutative reciprocity laws associated to finite groups